| L(s) = 1 | + (−9.97 + 1.75i)2-s + (24.1 + 12.1i)3-s + (36.1 − 13.1i)4-s + (60.9 − 72.6i)5-s + (−261. − 78.3i)6-s + (−133. − 48.6i)7-s + (223. − 129. i)8-s + (435. + 584. i)9-s + (−480. + 832. i)10-s + (1.62e3 + 1.93e3i)11-s + (1.03e3 + 120. i)12-s + (−126. + 717. i)13-s + (1.41e3 + 250. i)14-s + (2.35e3 − 1.01e3i)15-s + (−3.88e3 + 3.26e3i)16-s + (4.21e3 + 2.43e3i)17-s + ⋯ |
| L(s) = 1 | + (−1.24 + 0.219i)2-s + (0.893 + 0.448i)3-s + (0.565 − 0.205i)4-s + (0.487 − 0.581i)5-s + (−1.21 − 0.362i)6-s + (−0.390 − 0.141i)7-s + (0.436 − 0.252i)8-s + (0.597 + 0.801i)9-s + (−0.480 + 0.832i)10-s + (1.22 + 1.45i)11-s + (0.597 + 0.0696i)12-s + (−0.0575 + 0.326i)13-s + (0.517 + 0.0912i)14-s + (0.696 − 0.300i)15-s + (−0.949 + 0.796i)16-s + (0.856 + 0.494i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.596 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.08295 + 0.544511i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.08295 + 0.544511i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-24.1 - 12.1i)T \) |
| good | 2 | \( 1 + (9.97 - 1.75i)T + (60.1 - 21.8i)T^{2} \) |
| 5 | \( 1 + (-60.9 + 72.6i)T + (-2.71e3 - 1.53e4i)T^{2} \) |
| 7 | \( 1 + (133. + 48.6i)T + (9.01e4 + 7.56e4i)T^{2} \) |
| 11 | \( 1 + (-1.62e3 - 1.93e3i)T + (-3.07e5 + 1.74e6i)T^{2} \) |
| 13 | \( 1 + (126. - 717. i)T + (-4.53e6 - 1.65e6i)T^{2} \) |
| 17 | \( 1 + (-4.21e3 - 2.43e3i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-62.0 - 107. i)T + (-2.35e7 + 4.07e7i)T^{2} \) |
| 23 | \( 1 + (4.13e3 + 1.13e4i)T + (-1.13e8 + 9.51e7i)T^{2} \) |
| 29 | \( 1 + (-2.21e4 + 3.91e3i)T + (5.58e8 - 2.03e8i)T^{2} \) |
| 31 | \( 1 + (-1.41e4 + 5.13e3i)T + (6.79e8 - 5.70e8i)T^{2} \) |
| 37 | \( 1 + (-1.72e4 + 2.99e4i)T + (-1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + (8.70e4 + 1.53e4i)T + (4.46e9 + 1.62e9i)T^{2} \) |
| 43 | \( 1 + (4.95e3 - 4.15e3i)T + (1.09e9 - 6.22e9i)T^{2} \) |
| 47 | \( 1 + (-5.96e4 + 1.63e5i)T + (-8.25e9 - 6.92e9i)T^{2} \) |
| 53 | \( 1 - 2.77e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (-6.67e4 + 7.95e4i)T + (-7.32e9 - 4.15e10i)T^{2} \) |
| 61 | \( 1 + (2.84e5 + 1.03e5i)T + (3.94e10 + 3.31e10i)T^{2} \) |
| 67 | \( 1 + (7.05e3 - 3.99e4i)T + (-8.50e10 - 3.09e10i)T^{2} \) |
| 71 | \( 1 + (8.73e3 + 5.04e3i)T + (6.40e10 + 1.10e11i)T^{2} \) |
| 73 | \( 1 + (-1.59e5 - 2.75e5i)T + (-7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (-1.17e4 - 6.63e4i)T + (-2.28e11 + 8.31e10i)T^{2} \) |
| 83 | \( 1 + (9.75e5 - 1.72e5i)T + (3.07e11 - 1.11e11i)T^{2} \) |
| 89 | \( 1 + (1.69e5 - 9.79e4i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 + (-9.81e5 + 8.23e5i)T + (1.44e11 - 8.20e11i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.59143083879334815328348446552, −15.18990271789207063995756824233, −13.90011252943384485070535928477, −12.48695764975568683232251265263, −10.15671272888729288920391646092, −9.513563306340111874200109526233, −8.487845488228712791259574822730, −7.00903951116024956798215645228, −4.30239394815521649551556003621, −1.61269623752675205124917179934,
1.13650830990174735426884987872, 3.06567213644819312787242656600, 6.46417044341946819333415029701, 8.036402912990855935773208761640, 9.161037278793227408384916949424, 10.14009661499303150013821996697, 11.74726504697007505399474797805, 13.65139610327940066077269834879, 14.35079063271483588461751883541, 16.12586247732643564364489537441
